T-duality of curvature terms in D-brane actions

We examine the known curvature terms in the DBI part of the D-brane action under the T-duality transformation. Using the compatibility of the action with the standard rules of T-duality at the linear order as a guiding principle, we include the appropriate NS-NS $B$-field terms in the action and show that they reproduce the O(\alpha'^2) terms of the corresponding disk-level scattering amplitude.